Chapter 4 Partition
(1) Shifting
Ding-Zhu Du
Disk Covering
• Given a set of n points in the Euclidean
plane, find the minimum number of unit
disks to cover the n given points.
a
(x,x)
Partition P(x)
Construct Minimum Unit
Disk Cover in Each Cell
1/√2
Each square with edge length
1/√2 can be covered by a unit
disk.
Hence, each cell can be covered
By at most a 2 2 disks.
Suppose a cell contains ni points.
Then there are ni(ni-1) possible
positions for each disk.
Minimum cover
can be computed
2
)
In time nO(a
i
Solution S(x) associated with P(x)
For each cell, construct minimum cover.
S(x) is the union of those minimum covers.
Suppose n points are distributed into k cells
containing n1, …, nk points, respectively.
Then computing S(x) takes time
2
O(a )
n1
2
+ n2
O(a )
2
O(a )
+ ··· + nk
2
< n
O(a )
Approximation Algorithm
For x=0, -2, …, -(a-2), compute S(x).
Choose minimum one from S(0), S(-2), …, S(-a+2).
Analysis
• Consider a minimum cover.
• Modify it to satisfy the restriction, i.e.,
a union of disk covers each for a cell.
• To do such a modification, we need to
add some disks and estimate how many
added disks.
Added Disks
Count twice
Count four times
2
Shifting
2
Estimate # of added disks
Shifting
Estimate # of added disks
Vertical strips
Each disk appears
once.
Estimate # of added disks
Horizontal strips
Each disk
appears once.
Estimate # of added disks
# of added disks for P(0)
+ # of added disks for P(-2)
+ ···
+ # of added disks for P(-a+2)
< 3 opt
where opt is # of disk in a minimum cover.
There is a x such that
# of added disks for P(x) < (6/a) opt.
Performance Ratio
P.R. < 1 + 6/a < 1 + ε
when we choose a = 6 ⌠1/ε .
2
Running time is n.O(1/ε )
Unit disk graph
<1
Dominating set in unit disk graph
• Given a unit disk graph, find a dominating
set with the minimum cardinality.
• Theorem This problem has PTAS.
Connected Dominating Set
in Unit Disk Graph
• Given a unit disk graph G, find a minimum
connected dominating set in G.
Theorem There is a PTAS for connected
dominating set in unit disk graph.
central area
h+1
h
Boundary area
Why overlapping?
cds for G
cds for each
connected
component
1
Construct PTAS
For each partition P(a,a), construct C(a) as follows:
1. In each cell, construct MCDS for each connected
component in the inner area.
2. Connect those minimum connected dominating sets
with a part of 8-approximation lying in boundary area.
Choose smallest C(a) for a = 0, h+1, 2(h+1), ….
Existence of 8-approximation
1. There exists (1+ε)-approximation for minimum
dominating set in unit disk graph.
2. We can reduce one connected component with
two nodes.
Therefore, there exists 3(1+ε)-approximation for
mcds.
8-approximation
1. A maximal independent set has size at most
4 mcds +1.
2. There exists a maximal independent set,
connecting it into cds need at most 4mcds nodes.
MCDS (Time)
1. In a square of edge length 2 / 2 , any node can
dominate every bode in the square.
Therefore, minimum dominating set has size
at most (a 2 ) 2 .
a
2/2
MCDS (Time)
2. The total size of MCDSs for connected components
in an inner square area is at most 3(a 2 )3 .
a
2/2
Suppose a cell cotains ni nodes. Then finding
all MCDSs in the cell for all connected
components in the inner area takes time
ni
O ( a2 )
.
Over all cells, the total time is
n
i
i
O(a2 )
O(a2 )
n
MCDS (Size)
• Modify a mcds for G into MCDSs in each
cell.
• mcds(G): mcds for G
• mcdscell(inner): MCDS in a cell for
connected components in inner area
Connect & Charge
charge
Multiple Charge
How many possible
charges for each node?
charge
How many components
can each node be
adjacent to?
1. How many independent points can be packed
by a disk with radius 1?
5!
1
>1
Each node can be charged at most 10 times!!!
k nodes can connect to at most 5k components .
Each component make a charge to 2 nodes.
At most 10k changes will be mde on k nodes.
Each node receives at most 10 charges.
Shifting
h=2
a/(2(h+1)) = integer
2
Time=n O(a )
3
(1   ) - approximat ion in time n
O (1/  2 )
in any dimesion.
Weighted Dominating Set
• Given a unit disk graph with vertex weight,
find a dominating set with minimum total
weight.
• Can the partition technique be used for the
weighted dominating set problem?
Dominating Set in
Intersection Disk Graph
• An intersection disk graph is given by a set
of points (vertices) in the Euclidean plane,
each associated with a disk and an edge
exists between two points iff two disks
associated with them intersects.
• Can the partition technique be used for
dominating set in intersection disk graph?
Thanks, End